Finite element elastic-plastic analysis of cracks

0045-7949189 s3.00 + 0.00 Pergamon Press plc

Compurers & SI~UL.IU~PSVol.33. No. 2. pp. 363-373. 1989 Printed in Great Britain.

FINITE ELEMENT ELASTIC-PLASTIC ANALYSIS OF CRACKS A, N. PALAZOTTO and J. G. MERCER Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, U.S.A. (Received 26 Sepember 1988) Abstract-The finite element method is used to investigate a fatigue type loading. A crack closure algorithm is evaluated and discussed, This algorithm is implemented in the SNAP program and found to be satisfactory with linear or quadratic element formulations.

INTRODUCTION

Numerous efforts have been undertaken using elastic-plastic finite element analysis to characterize the behavior of cracks. A review of recent works will be covered. Newman [ 1,2] performed detailed analysis to study crack closure and crack growth. He adjusted the boundary conditions along the crack using spring elements and showed that a crack may ciose at a positive load, 40-75% of the maximum applied value. This was true for loading at positive and negative R-ratios, where the R-ratio is defined as the minimum applied load divided by the maximum applied load. He also examined crack growth by releasing crack tip nodes when a critical value of strain was reached, and achieved good results. Zahoor and Abou-Sayed [3] performed finite strain, elastic-plastic analysis on a center-cracked panel geometry. Constant triangles and four-noded quadrilateral elements were employed. They observed element rotation at the crack tip which resulted in its blunting. Altus [4] used finite element analysis with both small and finite strain fo~ulations to examine the strains and plastic energy density in the crack tip regions. The results showed little difference in J integral values for the two methods, but plastic energy density differed by almost 50%. Hammel et al. [S] performed elastic-plastic finite element modelling of a compact tension specimen under monotonic loading. They examined the effects of mesh size and small vs finite strain theory. They found the difference between the two formulations most pronounced at high loads where the finite strain results gave larger deflections and plastic zones. Yagawa et al. [6] described a round-robin exercise involving six teams performing elastic-plastic element modelling of a center-cracked specimen. Various element types were used as well as small and finite deformation theory. Similar results were found for the various parameters: displacement, J integral, plastic zones, and COD. The isoparametric element

formulations were shown to be in best agreement with experimental load vs displacement data. Hinnerichs [7] developed ‘VISCO’, a viscoplastic 2D finite element program. This program employed constant strain triangles, small strain formulation and various flow laws along with an Euler explicit time integration scheme. Hinnerichs examined constant load creep crack growth for a center-cracked specimen of IN-100 at 1350°F. He simulated crack extension by releasing nodes ahead of the crack tip. Using a hybrid experimental-numerical technique, he matched crack opening displacements and attempted to define a crack growth criterion. Damage accumulation and critical strain were found to be the most promising criteria. The work of Hinnerichs has been extended in recent years by Smail and Palazotto [S], Keck and Palazotto [9], and Wilson [lo]. These efforts included modelling new geometries and cyclic loading. Mercer and Palazotto [l I] discussed a 2D plane stress and plane strain program developed initially by Brockman [ 121 called SNAP. A major revision was carried out by these two authors, incorporating an ability to trace crack growth numerically under fatigue loading for materials within a viscoplastic regime. The subsequent paragraphs illustrate some of the major considerations implemented in this updated version of the SNAP code. FINITE ELEMENT

PROGRAM

A great deal of the theory in SNAP related to elastic-plastic and elastic-vicoplastic equations was given in [1 l] and these expressions will not be repeated here. The technique of solving the nonlinear equations was also presented with the appropriate tolerances. A center-cracked plate under a nonotonic load was a major consideration in [I I]. The authors considered large and small displacement theory as well as elastic-plastic and elastic-viscoplastic comparisons. We attempt here to extend the previous publication by considering additional 2D models as 363

364

A. N.

well as fatigue type loadings ation of crack extension and It is possible to express the and strain in an incremental

PALAZOTTO and

requiring the considerclosure. relation between stress form:

{dot = ]Dlf{dt I-

id@}],

(1)

where [1)] is the elastic stiffness matrix. It is further possible to determine the incremental strain functions as either time independent (elastic-plastic) or time dependent (plastic-viscoplastic). This could, in general, be expressed as i,,=i&-+C[,

(2)

where ii is the elastic strain rate which could be reduced to the elastic strain inurement considering time independency, and i& is the viscoplastic strain rate which could be effectively reduced to the Prandtl-Reuss relations for time independent plasticity. The program developed can consider each individuaIly with B~ner-Partom expressions [ 1I] as the general elastic-viscoplastic constitutive equations. The resulting equations require an incremental approach with a time step interaction dictated by the average strain rates for the beginning and end of an increment. This can be expressed for a given time step, n, as A,,,cr= Ar[(l - B),,,_&,,+ Q,];

(3)

for @= 0 we obtain the Euler integration scheme (fully explicit); for 8 = 1 we obtain a fully implicit scheme; 0 = 0.5 is known as the Crank-Nicolson rule or semi-implicit scheme. The program uses the implicit scheme in which the initial estimates of &onditions at the end of the increment are obtained using the Euler extrapolation. If necessary, an iteration loop is then used to recalculate stress values. The elastic-plastic and the viscoplastic models both employ a strain subincrementation scheme on an element~by-element basis. Within a load step, element incremental strains, Akt, at each Gauss inteI

1~

K

J. G.

MERCEK

gration point are further divided to allow a maximum component total strain of 2 x lo-‘, as illustrated in Fig. 1. Stresses and plastic strains are calculated for each subincrement, allowing the element to follow the stress-strain behavior closely for the appropriate constitutive model even when the load step is large. This operation is done for each Gauss integration point of each element independently, as the constitutive model is repeatedly called by the element integration routine. Therefore, the number of subincrements varies from point to point, depending upon the magnitude of the strain increment in each element. This provides for more efficient computation for the model as a whole. Crack growth and closure

Crack growth and closure have been implemented in the SNAP program by the addition of ‘springs’ to the nodal degrees of freedom along the crack line, similar to the method described by Newman [13]. To demonstrate this concept, consider a typical linear, 20 element located on the crack surface, as shown in Fig. 2. One-dimensional springs are attached to all nodes behind the crack tip, as well as nodes ahead of the crack tip over the distance the crack is to be grown. These springs have the capability to restrain the crack surface perpendicular to the direction of the crack (y-direction) while allowing freedom of movement along the crack plane (x-direction). The eiement stiffness has the form 1 %,

k,J...k,a

k>,

[j-q =

. . . k *,

kz2..

.

’ .” . .*... . ..I.

One-dimensional spring elements A and B are nodes 1 and 2. These springs restrain those nodes in the y-direction, and introduce two more degrees of

Subincrementel Strein c--

,

I’

,

If

I’ :

I’

+

------

cons~tutl~ law

_

Subincremental Stress-Strein Lew

A& A E E Incrementel Strain

Strain

Fig. I. Strain subincrementation

(4)

. .

,’

StnSS

*

for elastic-plastic and viscoplastic constitutive models.

365

Finite element elastic-plastic analysis of cracks

The spring stiffness, k,, remains only in those degrees of freedom corresponding to the y displacement of nodes 1 and 2. By making the spring very stiff, we simulate a fixed boundary, or crack closure. This stiffness is set to 1000 times the maximum diagonal entry in the structure stiffness matrix. We note here that if the crack is not to be grown during the course of an analysis, a fixed boundary condition is applied to the nodes ahead of the crack tip instead of the ‘springs’ discussed above.

Crack lip

IA-

X

+

Finite element implementation Fig. 2. Crack-tip finite element model. freedom: one each at nodes 5 and 6. The stiffness matrices for springs A and B are

(5) where k, represents the stiffness of the springs. The global stiffness matrix [K] is obtained by adding the element and spring stiffness matrices in accordance with their global degrees of freedom:

For those nodes lying along the crack line, a set of flags for each node behind and ahead of the crack tip is maintained. These are input data with a positive value indicating fixed, and negative value indicating a free node. Values of ‘1’ and ‘2’ are used to indicate x- and y-directions, respectively. Crack growth, or node release, is accomplished when a user-defined criterion such as stress is exceeded. This criterion may be any relevant quantity such as stress, strain or load. The flag corresponding to that nodal degree of freedom is set equal to a negative value and the structure stiffness recalculated without the spring stiffness at

(6) 1

2

3

4

5

k, , ..............

...........

3

k 3, ..............

...........

k4, ..............

...........

k 5, ..............

...........

6

k, , ..............

...........

7

k , , ..............

...........

k 8, . . . . . . . . . . . . . . .

..

L

k12 k,,

The numbers along the top and sides of the matrices represent the global degrees of freedom. Recalling that nodes 5 and 6 are restrained, those degrees of freedom are eliminated from the solution process, resulting in a stiffness matrix of the form:

k,, k,3 k,, k,, h, k,, k,, k,, kx k,, k,, k,,

k 33 (k,

WI = Symmetric

+

k3,

k35

k36

k3,

k3,

k)

h5

k6

b,

k,,

k 5s

k56

k5,

48

k,

k,,

k,,

k,,

k,, k 88

(8) c A.8

33,2--D

8

2

8

(k,, + k) kn

7

k,,

WI=:

4,

6

kM k,S k 16

1

h,

k,,

9 1 + k,

[ -1 10 1

+ k,

[ -1

2 -1 1 4 -1

I

1 9 2

(7)

1

10 4

. . k 88

that node. That particular load increment is then solved again with what is essentially a new boundary condition. Displacements of nodes behind the crack tip are checked for negative values after each load increment to determine if the crack has closed. If the node has negative displacement, its flag is set equal to a positive value, and the stiffness matrix reformulated with the spring added. Solution then proceeds with the next load increment. To minimize ‘overshoot’ in the negative direction, it was found that load steps less than 5% of maximum load were acceptable. The nodal reaction force at each ‘closed’ node is the negative of the internal force vector which is calculated using the usual surrounding node element technique as discussed in [14]. When this reaction force exceeds a preset value, the node is released and the load increment is reset, as for crack growth. The

A. N. PALAZOTTO and J. G. MERCER

366

Table 1. Bodner-Partom Bodner’s material parameters

parameters developed for IN- 100 In 100 1350°F (742°C)

Description Elastic modulus Strain rate exponent Limiting value of strain rate Initial value of hardness Maximum value of hardness Minimum value of hardness

m

Hardening rate exponent

A I

Hardening recovery coefficient Hardening recovery exponent

value for the opening force may be calculated using a small positive stress, such as 5 ksi, in a direction normal to the crack. For four-noded elements, 0.002in. long, this corresponds to a nodal force of 0.01 kips, considering contributions from two adjacent elements. For two adjacent linear elements with an edge length of 0.002 in. (0.05 mm), this corresponds to a nodal force of 0.01 kips (44.5 N). RESULTS AND

DISCUSSION

We will first consider a ID viscoplastic problem with a cyclic loading using IN-100 at 1350°F and the Bodner-Partom constitutive mode1 represented with the nine material constants as shown in Table 1. (Further discussion of this model can be found in [S]-[lo].) The 1D finite element model is that shown in Fig. 3. Cyclic loading was performed to a peak load of 200 kips at 0.167 Hz with an R ratio of - I .O (fully reversed), as depicted in Fig. 4. Loading was repeated for a total of five full cycles (R = min load/ max load). Time steps for loading were chosen based upon the results of monotonic loading to peak load. Beginning with a time step of 0.1 set during nonlinear deformation, the time step was repeatedly halved and the final strains compared. The difference in strains at maximum load calculated using a time step of 0.005 and 0.01 was only 0.3%, but CPU time was 50% higher. Therefore, the 0.01 set time step was used in the cyclic loading case. This corresponds to a load increment of 1.33 kips. In the linear region, time steps of 0.5 set

26.3 x lo3 ksi (1.8134 x IO5MPa) 07 lo4 set’ 915.0 ksi (6304 MPa) 1015.0 ksi (6993 MPa) 600.0 ksi (4134 MPa) 2.57 ksi-’ (0.3727 MPa-‘) 1.9 x 10-j see-’ 2.65

(66.7 kips) were used. This choice gave 104 load increments per load cycle or 1040 increments for the entire five cycles of loading. The modified Newton-Raphson solution algorithm, and displacement and force tolerances of 10m5 and 0.1, respectively, were used. Solution time was 175 set on the Cyber 845, or 26 set on the Cray XMP computer. Figure 5 shows the stress-strain history for the five cycles of loading using the implicit Bodner algorithm described previously and in [I 11. The results shown are average values for elements 3 and 4. The first stress-strain loop showed a positive plastic strain equal to 0.0028 in/in. Subsequent cycles were all found to be coincident with one another. Table 2 provides a summary of peak and maximum stress and strain for each cycle. This finite element analysis gave a final strain within 1% of a 1D numerical integration solution found in [15]. Thus, the SNAP finite element program viscoplastic routine was found to give acceptable results. V-Notched plate

The behavior of an elastic-perfectly plastic tension with a centrally located V-notch was examined. This problem was described by Yamada et al. [16] and the geometry and material properties are shown in Fig. 6. Because of symmetry, only one-quarter of the plate was modelled. Four different finite element meshes were used, as shown in Fig. 7. Two meshes (a and b) used linear displacement elements, whereas the remaining two meshes (c and d) used quadratic elements. Meshes a, b, and d had a like number of degrees of freedom (29&298), whereas mesh c had 518 degrees of freedom. It employed the same size tPLoad

L = 3.0 inches H I t =1.0 inches P=MOKIPS v = 0.0

Fig. 3. One-dimensional element model.

.P I 0

I

6

I

12

I

Time (set)

16

Fig. 4. Loading history, Bodner-Partom

I

24

I

30

ID model.

367

Finite element elastic-plastic analysis of cracks 300.00

\-

YlllAl LOADINO

(CYCLE 1)

200.00

_ lw.w

2 t R

.w

-100.00

-200.00

_.

-so

.w

110

1.50

1.00 PERCENT

Fig. 5. Bodner-Partom

2x4

2.50

3.00

STRAIN

ID model stress-strain

response.

Table 2. Stress-strain summary Cycle 1 Strain Stress (ksi) Max load l/2-cycle Min. load Full cycle

200.01 0 - 200.01 0

Cycle 2-5 Strain Stress (ksi)

0.017396 0.016653 0.002031 0.002801

200.01 0 - 200.01 0

Rajendran: Strain

0.017423 0.016652 0.00203 1 0.002801

0.017343 0.016658 0.002144 0.002829 Max strain

The maximum strain was 0.022886 in/in. at 172.0 ksi

elements as mesh b, but with eight-noded quadratic elements in the vicinity of the notch. Each model was loaded in 30 steps to 0,/a,, = 1.183, where aM is the mean stress at the notch location, given by a,,, = 2P/40 (kg/mm2). Additional load increments were then applied until a load of aM/a, = 1.1225 was attained or convergence was not obtained in 20 iterations, indicating that the structure was approaching collapse. Elastic solution

For an elastic material, the sharp notch introduces a stress singularity, which makes the stresses approach infinity at the notch root for any applied load.

0.022867 at 168ksi

In his work, Yamada showed initial yielding at the notch root to occur at a load of a,/a,, = 0.370. In this paper, however, the effective stress remained below yield at this load for all four element meshes, so no plastic deformation was observed. The elastic axial stress, a,, , at the notch is shown as the bottom set of curves in Fig. 8. All stresses shown are at the element integration points nearest to the plate centerline. Stresses are shown to have a similar trend for all four meshes. The quadratic elements (meshes c and d) appear to accommodate the large stress gradient better than the linear quadrilaterals (mesh b) and constant strain triangles (mesh a). Very little difference is seen between the coarse and fine quadratic element solutions at this load level. Elastic-plastic solution

Pohwn’~ Iwoi 0.30 VIM SIN9(a,,dok#mm Fig. 6. V-Notched plate geometry.

Stresses. The elastic plastic analysis was performed using the constant (initial) stiffness approach and an elastic-perfectly plastic material model. Stresses were examined at a load level of by/a,, = 1.17, for which plastic deformation was present; and are shown in Fig. 8. The fine quadratic element mesh (mesh c) gave axial stresses (a,) relatively constant throughout the yielded region, followed by a reduction of stresses in the elastic region away from the notch. The other meshes showed a similar trend away from the notch root, but with a lower stress level at the notch itself.

A. N. PALAZOT~O and J. G. MERCER

368 MESH a 272 ELEMENTS 162 NODES

MESH d 45 ELEMENTS

(LINEAR)

(alJADRATIC)

164 NODES

297 DOF

298WF

MESH b 136 ELEHEN-TS

I -

(LINEAR)

161 NODES 296OOF

4

MESH c 138 ELEMENTS 280 NODES 518 DOF

a= ELEMENTS

NOTCH PLANE

(aUADRATlC

AND

I

Fig. 7. V-Notched plate finite element model.

The coarse quadratic mesh (mesh d) gave higher stresses than the other meshes away from the notch. The trends shown in Fig. 8 are also similar to those reported by Nayak and Zienkiewicz [ 171 for various mesh arrangements. Plastic zone. Yield zones are shown in Fig. 9 for loads of cry/a,, = 1.183 and the highest applied load. The extent of the plastic zone was identified by examining the effective stress at each Gauss point for quadrilateral elements, or the single-element value for the triangular elements. Yielding started at the notch root, and extended diagonally toward the plate central axis (x-axis). At a load aM/a,, = 1.183, meshes b and c show nearly identical plastic zones. The triangular mesh (mesh a) had a plastic zone

which extended to the centerline (x-axis). For all meshes, as the applied load was increased above a,+,/asS= 1.17, the stresses at the plate centerline approached yielding; this resulted in a plastic zone which extended diagonally from the notch root to the plate centerline. Displacement. Figure 10 shows displacement vs load for the plate centerline at the edge where the load was applied (point c). For each mesh, the load was applied in the same number of increments. Intermediate displacements are shown only for mesh a. With the exception of mesh d, the present work shows the plate to be slightly stiffer than in Yamada’s results. One possible explanation for this difference is that Yamada used 0.995a,., as the point

A MESH b 0 MESH c 0 MESH d

L---l

10.00

15.M DISTANCE

2D.00 FROM NOTCH

25.00 (NM)

Fig. 8. Axial stresses at the notch.

3m

35.00

Finite element elastic-plastic analysis of cracks

MESH(b)

D

NOTCH’PLANE

369

MESH (c) 0

Fig. 9. V-Notched plate plastic zones.

for calculation than a,,.

of the elastic-plastic

stiffness rather

Parameter variation. The effect of the number of load increments and integration points on the solution was examined. The V-notched plate was analyzed using the coarsest mesh with 30 and 15 increments of load to tr,,&,, = 1.183 and four- and nine-point integration. The solution was found to be insensitive to these changes in solution parameters. The plastic zone sizes were found to be virtually identical to those of Fig. 9 for all four cases. The axial stresses shown in Fig. 11 showed only small differences in the unyielded region away from the notch. Solution time. For the four meshes examined, the following CPU times on the Cyber 845 computer were required for loading to ~,JQ, = 1.183 in 30 increments.

CPU Mesh a Mesh b Mesh c Mesh d

Mesh

time

(CST) (four-noded quadrilaterals, four-point integration) (four- and eight-noded quadrilaterals, coarse mesh, four-point integration) (eight-noded quadrilaterals, coarse mesh, four-noint integration)

89 set 101 set 102 set 41 set

The coarse, quadratic element (eight-noded) required the lowest computation time. The stresses at the plane of the notch were similar to the other three meshes, but the plastic zone at load ~~/a,, = 1.183 was slightly larger. Looking further at the coarse eight-noded mesh (mesh d), the computation time for a different number of load increments and four- and nine-point integration orders was found as shown below: Mesh Mesh Mesh Mesh

d d d d

30 increments, four-point integration 30 increments, nine-point integration 15

increments, four-point integration

15 increments, nine-point integration

41 set 17 set 28.8 set 45 set

The nine-point integration needed almost double the computation time of the four-point integration. Presumably this is because stresses and strains are evaluated at roughly twice the number of points in an element. The larger load steps (15 increments) also reduced computation time considerably ( z 60%) without any appreciable change in results (stress or plastic zone size). Thus, it is advantageous to use the four-point integration and as large a load step as is allowable. Additionally, the eight-noded elements were found to provide an acceptable answer using a considerably coarser mesh. Cyclic loading. A center-cracked panel shown in Fig. 12 (containing 488 DOF with a crack tip element of 0.0015 in. using an eight-noded quadratic) was subjected to cyclic loading at 2.5 Hz, R = - 1.0 as depicted in Fig. 13. Crack-opening displacement at minimum load (0.3 set) is shown in Fig. 14. Two items are to be noted with regard to this profile. First, crack closure prevents crack surface displacement below the line of symmetry. Second, this opening profile represents only the stationary crack and is not representative of a crack which has grown and has a plastic wake. As such, this profile serves only as a check of the finite element crack closure routine for the quadratic element edges. The stress profile (crJ ahead of the crack tip is shown at 0.1 and 0.9 set (1.15 and 2.25 cycles, respectively) in Fig. 15. The inflection point in the upper curve (also observed in [18] using a constant strain triangle and the VISCO code), approximately 0.003 in. from the crack tip, is seen in all three solutions and corresponds to the zone of reversed plasticity under negative loading. Reversed plasticity refers to the compressive yielding which occurs near the crack tip as the applied load is reduced. The extent of this reversed plastic zone is also seen in the lower curves of Figs 15 and 16. The reversed plasticity is responsible for the hysteresis loops shown for the element nearest the crack tip in Fig. 17. Stress-strain response (a,,) for the element integration point nearest the crack tip is shown in Fig. 17.

370

A. N.

PALAZOTTO and J. G. MERCER

0 MESH

ci

AYESH b OYESH OMESH

c d

[lb]

_Yamada

.02

.04

06 DISPLACEMENT

.06 (MM)

Fig. 10. V-Notched plate load&placement

We see in this plot that the majority of strain is accumulated in the first cycle of loading. Because of crack closure, the strains remain positive even at full negative applied load. Without considering closure, the stress-strain response would be nearly symmetrical about the origin with a strain range of about -6% to +6% (Fig. 18) or three times larger than with the closure algorithm. This shows the importance of modelling closure with respect to the resulting response at the crack tip. CONCLUSIONS

Applying the SNAP program to the various problems discussed provided important information which is summarized below. (1) The SNAP program was found to provide satisfactory solutions for a wide range of problems

.12

.lO

.14

response.

and constitutive models. In the 1D problem the Bodner implicit algorithm provided an accurate solution. For center-cracked panel and compact tension specimen geometries, the results published by Henkel and Palazotto [18] were repeated satisfactorily. Elastic and elastic-plastic analysis of the V-notched plate essentially duplicated results reported by Yamada et al. [16]. (2) Eight-noded quadratic elements were applied to the V-notched plate and center-cracked panel problems. Larger elements may be used than when using constant-strain triangles. The eight-noded elements permit linear strain variations, which better accommodates high strain gradients. For the center-cracked panel it was found that the eight-noded element also has the ability to model curved surfaces, because of its quadratic shape functions

1.25 0 4 PT INT. 0 9 PT INT.

,

15 INCR. 30 INCR.

MESH d

1.00

9 F

0.75

s 0 3 3I

0.50

0.25

0

5.00

10.00

15.00 DISTANCE

20.00

25.60

FROM NOTCH (MM)

Fig. 1I. Axial stresses at notch-parameter

variation.

96.60

t

3.5.w

371

Finite element elastic-plastic analysis of cracks

, f‘it?

ETI

32o.w

0 225 CYCLES ( YAK LOAC ) A 1.75 CYCLES ( YIN LOAD Rz.l.0

240.00

16O.W

-240.M

Dimensions in ( inches

.w

am

10.00

40.06

3o.w

DISTANCE FROM CRACK ( IN’ lo”)

)’

Fig. 15. CCP stress profiles, cyclic loading maximum and minimum loads.

Fig. 12. Center-cracked panel finite element model

lP Load -P I

I

0

0.2

I

0.8

0.6

PLASTIC

70.04

I

I

I

0.4

STRAIN

1.0 0 0.25 CYCLES ( MAX LOAD ) 0 0.75 CYCLES ( MIN LOAD ) R z.1.0

Fig. 13. CCP cyclic loading history. mm

r 1.25

35KSIfi 2.5HZ COD AT MIN LOAD

LCCP

55.55

( 0.75 CYCLES) R : -1.0

1.00

40.00

8

5 % k

Pm w

0.7!

fi

300 D-

! 2 1s 0.5t

1 Y ,I

I 20.0 0'

THE DIFFERENCE BETWEEN THESE CURVES REPRESENTS COMPRESSIVE YiELDiN

‘$T

REVISED PLASllCZONE

10.0 0

0.24

I 4o.w

m

O

I

”

-30.00 DISTANCE

.2mo

.w

BEHIND CRACK (IN ’ 10’ )

Fig. 14. CCP crack-opening displacement at minimum load.

1)0 Do

10.00

25.00

5&W

45.44

DISTANCE FROM CRACK (MM)

Fig. 16. CCP plastic strain profiles, cyclic loading maxrmum and minimum .loads.

372

A. N. PALAZOTTO and J. G. MERCER bOO.Ot

L

STRESSSTRAIN AT THIS POINT

COMPRESSIVE

YIELDING

6

I

10.06

20.00

40.66

36.06

50.06

6tJ.06

1

70.00

STRAIN E, (IN/ IN ’ 10’)

Fig. 17. CCP cyclic stress-strain

response.

j WI0

LOOP IS 3 TMES

400.00

LARGER

1

M1.00

.26.lM

1

THAN WITH CLOSURE

I 40.66

CLOSURE

20.00

.66

40.66

I

66.66

1 66m

STRAIN EY (INI IN’ 10’)

Fig. 18. Cyclic stress-strain

response without closure algorithm.

(3) The crack closure algorithm implemented in the SNAP program was found to work satisfactorily with linear or quadratic element formulations. (4) During the loading, most of the plastic strain is accumulated in the first cycle of loading. This was true for the CCP undergoing fully reversed loading V = - 1.0) and the CT specimen with R = 0.1. REFERENCES J. C. Newman, A finite analysis of fatigue crack closure. American Society Testing and Materials, ASTM-STP590, 281-301 (1976). J. C. Newman, Finite element of crack growth under monotonic and cyclic loading. American Society for Testing and Materials, ASTM-STP-637, 55-80 (1977). A. Zahoor and I. S. Abou-Sayed, Prediction of stable

4. 5. 6.

7.

8.

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